[Paper Review] The Vanishing Moment Method for Fully Nonlinear Second Order Partial Differential Equations: Formulation, Theory, and Numerical Analysis

Xiaobing Feng, Michael Neilan|arXiv (Cornell University)|Sep 6, 2011

Advanced Numerical Methods in Computational Mathematics59 references12 citations

TL;DR

This paper introduces the vanishing moment method as a novel numerical approach for solving fully nonlinear second-order PDEs, such as the Monge-Ampère, prescribed Gauss curvature, and infinity-Laplacian equations. By regularizing the original PDE into a family of fourth-order quasilinear equations parameterized by ε, the method enables stable finite element and mixed finite element discretizations, with convergence to viscosity solutions proven under radial symmetry and validated numerically with optimal error rates.

ABSTRACT

The vanishing moment method was introduced by the authors in [37] as a reliable methodology for computing viscosity solutions of fully nonlinear second order partial differential equations (PDEs), in particular, using Galerkin-type numerical methods such as finite element methods, spectral methods, and discontinuous Galerkin methods, a task which has not been practicable in the past. The crux of the vanishing moment method is the simple idea of approximating a fully nonlinear second order PDE by a family (parametrized by a small parameter $\vepsi$) of quasilinear higher order (in particular, fourth order) PDEs. The primary objectives of this book are to present a detailed convergent analysis for the method in the radial symmetric case and to carry out a comprehensive finite element numerical analysis for the vanishing moment equations (i.e., the regularized fourth order PDEs). Abstract methodological and convergence analysis frameworks of conforming finite element methods and mixed finite element methods are first developed for fully nonlinear second order PDEs in general settings. The abstract frameworks are then applied to three prototypical nonlinear equations, namely, the Monge-Ampère equation, the equation of prescribed Gauss curvature, and the infinity-Laplacian equation. Numerical experiments are also presented for each problem to validate the theoretical error estimate results and to gauge the efficiency of the proposed numerical methods and the vanishing moment methodology.

Motivation & Objective

• To develop a reliable numerical framework for computing viscosity solutions of fully nonlinear second-order PDEs, which have historically resisted standard Galerkin methods.
• To establish a rigorous convergence theory for the vanishing moment method in the radial symmetric case, ensuring the regularized solutions approach the true viscosity solution as ε → 0.
• To develop abstract finite element convergence frameworks—both conforming and mixed—for general fully nonlinear PDEs, applicable to prototypical equations like Monge-Ampère and infinity-Laplacian.
• To validate the theoretical error estimates through comprehensive numerical experiments on three benchmark nonlinear PDEs with reported convergence rates.
• To explore practical implementation challenges, including nonlinear solvers and preconditioning strategies for large-scale simulations.

Proposed method

• The method regularizes a fully nonlinear second-order PDE by introducing a small parameter ε, transforming it into a fourth-order quasilinear PDE: εΔ²u + F(D²u, ∇u, u, x) = 0.
• The solution of the regularized problem is shown to converge to the viscosity solution of the original PDE as ε → 0, under appropriate structural assumptions on F.
• Conforming and mixed finite element methods are formulated for the regularized fourth-order equations, with linearization via Newton’s method applied to the resulting nonlinear algebraic systems.
• A multi-resolution or homotopy strategy is employed to generate effective initial guesses for Newton’s method, starting from larger ε and progressively refining to smaller ε.
• The convergence analysis is conducted in abstract settings, with error estimates derived for both conforming and mixed finite element approximations.
• Numerical experiments use ILU-preconditioned Newton solvers, with convergence rates compared to theoretical predictions.

Experimental results

Research questions

• RQ1 Can the vanishing moment method provide a stable and convergent numerical scheme for fully nonlinear second-order PDEs using standard Galerkin methods?
• RQ2 What is the convergence rate of the vanishing moment method in the radial symmetric case, and how does it depend on the regularity of the viscosity solution?
• RQ3 How do conforming and mixed finite element methods perform in approximating the regularized fourth-order PDEs, and what error bounds can be established?
• RQ4 Can the method be extended to time-dependent or parabolic fully nonlinear PDEs, and what are the challenges in such generalizations?
• RQ5 What are the most effective nonlinear solvers and preconditioning strategies for solving the large-scale nonlinear systems arising from the method?

Key findings

• The vanishing moment method successfully regularizes fully nonlinear second-order PDEs into fourth-order quasilinear equations, enabling the use of standard finite element methods.
• Convergence of the regularized solutions to the viscosity solution of the original PDE is rigorously proven in the radial symmetric case, with convergence rates established under sufficient regularity.
• Optimal convergence rates of O(h²) are numerically observed for the conforming finite element method and O(h) for the mixed finite element method in the Monge-Ampère and prescribed Gauss curvature problems.
• The method is numerically validated on three prototypical equations, with computed convergence rates matching theoretical predictions.
• The multi-resolution strategy for generating initial guesses significantly improves convergence robustness of Newton’s method in practice.
• Open problems include extending convergence theory to general F, developing the method for parabolic fully nonlinear PDEs, and handling degenerate or non-elliptic systems.